Operator-based quantum thermodynamic uncertainty relations

Abstract: The Heisenberg uncertainty relation, which links the uncertainties of the position and momentum of a particle, has an important footprint on the quantum behavior of a physical system. Motivated by this principle, we propose that thermodynamic currents associated with work, heat, and internal energy are described by well-defined Hermitian operators; i.e., we associate physical observables to quantum thermodynamic flows. The observables are defined such that their expectation values match the average values of the associated currents. These rates, or currents, differ from their classical counterparts due to the non-commutativity of the corresponding operators. Using the Robertson-Schr\"odinger uncertainty relation, we then obtain various thermodynamic uncertainty relationships between them. In particular, we connect the fluctuations in heat rate and thermodynamic power with those in internal energy. We further illustrate this approach by applying it to quantum batteries, where we derive an energy-power uncertainty relationship and show how measurements affect the fluctuations.